It seems a bit strange to have initialized properties when the object is actually not initialized?

Perhaps uninitialized isn't the right word. The error basically happens because you're reading a the isEmptyElement property before actually reading anything. The fact that other properties are able to be read is perhaps a bit weird (except for properties like depth maybe) but changing that would be a painful BC break.

Ah okay, yeah I think the error wording maybe should be improved then, as I thought it was similar to a DateTime object not being properly initialized

What about something like "Failed to read property because no XML data has been read yet" ?

I think that is indeed a better error message! :)